Control systems have been widely used in a variety of applications, such as, for example, industrial manufacturing applications, navigational systems, communication systems, and information storage and retrieval systems. Regulating control systems usually maintain a plant or actuator in a given state (i.e. position, velocity, acceleration, temperature, pressure, or pH) despite unknown external inputs which may tend to force the plant or actuator to drift away from the selected state. The control system may also cause the plant or actuator to change from one state to another state in a rapid and predictable manner.
One method of minimizing the time to change a second order plant from an initial state to a final state is to apply a maximum control signal (i.e. current, voltage, torque or pressure) to the plant to change the state as fast as possible to a halfway point. The time required to reach this halfway point may be one-half the time required to move from the initial state to the final state. At this point, an opposite control signal is applied to the plant to change the state of the plant to the final state. This is referred to as a "bang-bang" technique. The first phase is referred to as the first bang, while the second phase is referred to as the second bang.
However, variations in the control signals, the plant parameters, the environment surrounding the plant, and the mechanical characteristics of the plant can cause the plant to overshoot or undershoot the final state. Further, in discretized systems, the control inputs are typically changed at uniformly spaced time intervals (i.e. the update rate) and the state of the plant can usually only be measured at uniformly spaced points in time (i.e., the sampling rate). Assuming that maximum control signals are used, the probability of a required switching time coinciding with the update rate is very low. As a result, the maximal command signals are usually applied for either a longer or shorter period of time than is required. This will result in an error in the final state after the minimum number of discrete sampling intervals.